多复变Siegel半空间中多次调和函数的积分表示、增长性及其应用

本项目拟研究多复变Siegel上半空间中具有重要理论意义的L^p(1≤p≤∞)函数的Poisson-Szegö积分的增长性;和具有最小多调和控制的多次调和函数的积分表示和渐进性新问题，探索多重位势理论中的新思路。本项目还拟研究R^n的上半空间中具有紧支撑的非平凡L^2函数的Fourier变换的唯一性和增长性问题；以及应用函数论的综合理论到金融统计中，探索自适应Fourier分解方法和加窗的分数阶Fourier变换方法研究金融时间序列高频数据的波动率估计问题的数学理论并进行实证分析，进而将自适应Fourier分解和加窗的分数阶Fourier变换应用到欧式期权定价理论，针对高频数据进行理论证明和实证分析与比较。这是一个有着深度应用背景的函数论基础理论研究，研究内容新颖，交叉性强，不仅对于进一步揭示单复变与多复变之间的本质差别有重要的理论意义，而且还有实际应用，属于学科前沿的难点和热点问题。

This project intends to study the growth of Poisson-Szegö integral of L^p(1≤p≤∞)functions with important theoretical significance in several complex Siegel half space, and the new problems of integral representation and asymptotic property of plurisubharmonic function with the least polyharmonic majorant, and explore new ideas in multi-potential theory. Besides, the uniqueness and growth of Fourier transformation of L^2 nontrival function with compact support in the upper half spaec of R^n are studied as well.This project will also apply the comprehensive theory of functional theory to Financial Statistics, explore the mathematical theories and empirical researchs of the Adaptive Fourier Decomposition method and Windowed Fractional Fourier Transform for volatility estimation of high-frequency time series data, and then deal with the theoretical and empirical analysis of the Adaptive Fourier Decomposition method and Windowed Fractional Fourier Transform for European Option Pricing as well. This is a basic theoretical research with a deep application background,the contents are novel, cross-cutting . It is not only of big theoretical significance for further revealing the essential difference between simple and multiple complex variables，but also applicable. The combination of theory and application is a frontier issue.